Week 5

Question 1

Polynomial

Answer 1

p(x) = ax^n + bx^n−1 + · · · + cx + d

Question 2

Fundamental theorem of algebra

Answer 2

A polynomial of order n has exactly n roots. Comes from every polynomial has at least one root, as every polnomial can be expressed as linears and smaller polynomials down the chain.

Question 3

Complex pairs

Answer 3

If a polynomial has a complex root, the complex conjugate must also be a root. Hence polynomials with uneven orders must at least have one real root.

Question 4

turning roots back into a polynomial

Answer 4

for a polynomial in the form x^n + ax^n-1 + bx^n-2

a = sum of roots
b = sum of pairs of roots (ab + ac + bc for a cubic)
c = sum of triples …. etc

Question 5

Chain rule

Answer 5

F(x) = f(g(x)) int F’(x) = f’(g(x))*g’(x)

Question 6

Chain rule for functions of multiple compositions

Answer 6

for a function F(x) of the form a(b(c(x))) chain rule can be used by starting from the outside out.
a’(b(c(x))) * d/dx(b(c(x)) then do chain rule again for second part

Question 7

find f’(pi) if f(x) = sin(sin(sin(x)))

Answer 7

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